Pattern inventory ($F_G(x)={\sum }_{\text{coloring orbits}}{\prod }_{s\in S}{x}_{f(s)}$)

This is the sum of content monomials of colorings up to the group action. It equals the cycle index specialized at $t_r=p_r(x)$. One may first use finitely many colors; stabilization defines the corresponding homogeneous symmetric function. Calling the specialized cycle index a cycle indicator makes $Z_G=F_G$ an equality in the same variables.